Scheduling Combination and Headway Optimization of Bus Rapid Transit

Transcription

1 JOURAL OF TRASPORTATIO SYSTEMS EGIEERIG AD IFORMATIO TECHOLOGY Voume 8, Issue 5, October 28 Onine Engish edition of the Chinese anguage ourna Cite this artice as: J Transpn Sys Eng & IT, 28, 8(5), RESEARCH PAPER Scheduing Combination and Headway Optimization of Bus Rapid Transit SU Chuaniao, ZHOU Wei*, WAG Yuanqing Highway Coege of Chang an University, Xi an 7164, China Abstract: The fexibiity of bus rapid transit (BRT) in scheduing is one of the greatest differences with traditiona buses. In order to improve BRT operation quaity, the paper studied the headway optimization and scheduing combination of BRT vehices. A mode has been estabished to minimize passengers trave costs and vehices operation cost, and constraints incuded passenger voume, time, and frequency. The scheduing combination was composed by norma, zone, and express scheduing. The mode was soved by genetic agorithm of variabe-ength coding. The resut of the numerica case shows that: the optimization resuts can save 69.92% cost. The sensitivity anaysis shows that, under higher traffic voume or ower speed, the trave cost can be reduced through reasonabe scheduing combination. The method has been proved scientificay and is feasibe. Key Words: transit operation; bus rapid transit; scheduing; genetic agorithm 1 Introduction Traditiona bus scheduing is a kind of fixed scheduing, which is checked on the terminas. Because of the congestion in urban city, pubic transport vehices often arrive at station uneveny, which eads to instabe quaity and ow attraction. BRT is constructed by excusive anes and inteigent transportation systems. Moreover, BRT can provide scheduing combination to meet passengers trave demand we and reduce vehices operation costs. Most researches focused on traditiona bus scheduing, such as determining the frequency by genetic agorithms or mixed agorithm [1 6]. Teng and Yang studied bus frequency under the APTS [7], which did not consider the characteristics of BRT scheduing. Zou studied regiona scheduing with mixed scheduing but did not offer an agorithm [8]. Bai et a. optimized bus frequency by taboo agorithms [9], with no studies on combinations. Therefore, it is necessary to study the scheduing combination and headway optimization of BRT. 2 Scheduing forms of BRT Scheduing combination is the specia characteristics of BRT. According to vehice operation form and stops number, the scheduing is divided into norma scheduing, zone scheduing, and express scheduing, and so on. orma scheduing: vehices run aong the routes and stop at every station from the initia stop to the end. The vehice must run at fixed stations and compete the whoe routes, as shown in Fig. 1(a). Zone scheduing is defined as vehices ony run on high-traffic voume section or zone (Fig. 1(b)). Express scheduing, that is, vehices ony stop at certain station with arge passenger voume (Fig. 1(c)). In traditiona bus scheduing, the norma scheduing is the most popuar form. Whereas in BRT operation, there are scheduing combinations, which are more in accordance with the trips needed on the corridor than traditiona bus. (a) (b) (c) Fig. 1 (a) Express schedue; (b) norma schedue; (c) zone schedue Received date: Apr 11, 28; Revised date: June 26, 28; Accepted date: Juy 8, 28 *Corresponding author. E-mai: zhw59@hotmai.com Copyright 28, China Association for Science and Technoogy. Eectronic version pubished by Esevier Limited. A rights reserved.

2 SU Chuaniao et a. / J Transpn Sys Eng & IT, 28, 8(5), Optimization mode of BRT scheduing 3.1 Probem description and assumption This study focuses on BRT headway optimization and scheduing combination (norma, zone, and express) in a certain period, which examines the appropriate departure frequency and scheduing combination to minimize the obective function. First, the foowing assumptions are given: (a) BRT vehices run at constant speed, namey, the running time between stations is certain. (b) In the study period, the departure frequency is uniform. (c) The passenger arriva rate is uniform and unchanged in the given period. (d) The time to open and cose vehice doors are fixed. (e) There are enough vehices in every feet. 3.2 Symbo definition In the study, the key variabes are defined as foows: i BRT vehice, i 1, 2,,s ; stops on BRT routes, 1, 2,, ; scheduing form, =1 means norma scheduing, =2 is zone scheduing, =3 represents express scheduing; 1 the number of norma scheduing; 2 the number of zone scheduing; 3 the number of express scheduing; s the tota number of scheduing; d departure time of vehice i at stop ; a the arriva time of vehice i at stop ; h headway between vehice i 1 and i at stop ; t vehice running time between stop 1 and ; h fixed headway; T dweing time at stop; c acceeration and deceeration time; r,k passenger arriva rate from stop k to, 1 k ; r arriva rate at stop ; r r, k k1 A the number of aighting passengers at stop from vehice i; B the number of boarding passengers at stop from vehice i; L passenger number on bus when vehice i eaves stop ; W k the number of passenger from stop to k on vehice i; s k passenger number for stop k when vehice i eaves ; S the number of passengers on vehice i eaves stop ; -1 variabe, for the scheduing form, when vehice i stops at, the vaue is 1, otherwise is ; k -1 variabe, for scheduing form, vehice i stops at both and k, the vaue is 1, otherwise is ; i scheduing form of vehice i; C 1 vaue of passenger waiting cost (yuan/min); C 2 vaue of passenger on board cost (yuan/min); C 3 operation cost of vehices (yuan/min); T studied period. 3.3 Mode formuation Obective function min Z f f f s r h 1 1 i1, i1 1 2 f C S h s 2 2 ( 1 ) ( ) i1 1 f C L t c L A T s 3 3 ( -1 ) i1 1 f C t c T The obective function consists of the passenger waiting cost, the passenger on board cost, and the vehice operation cost. The passenger waiting cost incudes the average passenger waiting cost and the skipping-station waiting cost. The former is cacuated by the arriva rate at stop, and r mutipies the headway h and haf of the headway. The atter is cacuated by the skipped passenger of the former vehice S i 1, mutipied by the waiting time h. The passenger on board cost is divided into the passenger on board time and the dweing cost. The former is cacuated by vehice number mutipied by the running time. The running time incudes the interva running time and the two consecutive dweing times ( i, i, 1 ) c. The dweing time is obtained by the on board number minus the aighting number mutipied by the dweing time T. The vehice operation cost consists of the running time t, the cosing and opening time of two consecutive station ( ), and the dweing time i, T if present. 1 c Decision variabes are h and i (=1, 2, 3), which determine headway and scheduing form. Athough i is not shown in the obective function, it is directy reated to Constraints (1) Passenger number constraints The passenger number on board of vehice i at stop equas to vehice i and eaves 1 pus boarding passengers at stop minus aighting passenger at stop : L =L 1 +B A. The aighting passenger number of vehice i at stop equas to passenger number of the vehice i at a the former 1 station to the stop k mutipied by the vehice stop or not: A k 1 W k k 1 The boarding passenger number of vehice i at stop equas to passenger number of vehice i boarding at stop whie aighting at a the downstream stations mutipied by vehice stop or not: B W k k k1 The transferring passenger number from stop to stop k equas to the passenger number of former vehice i 1 and eaves stop to k pus the arriva passenger number in the waiting time. The arriva passenger number is cacuated by the arriva rate r, k mutipied by the headway h i, :

3 SU Chuaniao et a. / J Transpn Sys Eng & IT, 28, 8(5), 6167 W s r h k i1, k, k The eft passenger number of vehice i from stop to k equas to the passenger number eft by the former vehice pus the arriva passenger in the waiting time. The second part determines when the vehice stops both at stops and k, and the eft number is added; otherwise, there is no eft passenger number: s ksi1, k(1 k) r, kh (1 k) The eft tota passenger number of vehice i at stop equas to the sum of a the passengers boarding at stop aighting at the rest stops: S s k k1 (2) Time constraint The arriva time of vehice i at stop is equivaent to the departure time of vehice i at stop 1 pus running operation time, even if the dweing time is stop at and 1 or not: a d t c 1 1 The departure time of vehice i at stop is equivaent to the arriva time pus the dweing time. If it stops at the station, the dweing time is considered; otherwise, the dweing time is : d a T The headway of vehice i at stop is equivaent to the headway difference between vehice i and i 1 at stop : h d di 1, Headways of a vehices at the terminas are equivaent to the uniform headway: h i, 1 = h i+1, 1 =h. The headway of vehice i at stop equas to the uniform headway pus the dweing time on the former two stations: ( ) ( ) h h T c T c k i 1, k k1 k1 With uniform headway the studied period is divided into s headways: s h=t. (3) Frequency constraints When the service eve and bus anes capacity is considered, the bus headway needs restriction: h min hh max Boundary condition The boundary condition makes a significant impact on the resut. There are severa conditions needed to be carified: the ast vehice at ast stop is defined to the arriva time, that is d M, =a M,. The first vehice at the first stop equas to the departure time. a 1,1 =d 1,1. The headway of first vehice is defined as h 1, =. The first running time t is defined as t 1 =. 4 Genetic agorithm based soutions Genetic agorithm is a random search agorithm, which traces its roots to bioogica evoution. The agorithm is first proposed by professor Hoand, the University of Michigan, in Genetic agorithm is a highy efficient, parae, and goba search method, which has been widey used in combinatoria optimization, machine earning, signa processing, adaptive contro, artificia ife, and so on. The maor steps of genetic agorithm incude: parameter choice and initiaization, fitness vaue and genetic operator, and so on. 4.1 Parameters choice and Initiaization (1) Coding Coding is the key point in the mode soution because the mode contains a variety of variabes, incuding the headway and scheduing form combination. Moreover, the decisionmaking variabes are interreated and the ength is variabe. How to determine coding and the code ength is the specia probem in the soution. Modes incude the maximum and minimum headway, which determines the ength of coding interva. This soution is encoded by the variabe-ength coding genetic agorithm. To encode the scheduing form, the ength of the scheduing form is bus frequency. For three different scheduing forms, the corresponding 1 is represented as norma scheduing, 1 as zone scheduing, and 11 as express scheduing. According to the coding ength interva, three different modes are produced. For exampe, when the headway interva is [3, 15] and we take 5, the possibe coding is [ ], which gives one of the scheduing combination. (2) Initia popuation The initia popuation is randomy generated. The first term is to determine the popuation size ( chromosomes). The second is to randomy seect points as the initia soution in optimizing space. For exampe, [1, 3] is randomy generated a initia popuation, such as a 2 3 popuation, in which popuation size is 2 and the frequency number is 3. (3) Parameters choice The crossing rate P c and the mutation rate P m shoud be identified. 4.2 Fitness vaue cacuation In the proposed mode, the obective function is to minimize the vaue. The simpe fitness function is utiized here, namey, F(x)=M f(x). F(x) is the fitness function, f(x) is the obective function, and M is a arge enough constant. Due to arge numbers of parameters are invoved, it is difficut to cacuate the fitness function. With coding, known arriva rate, and the number of rest passengers on initia vehices, we get the reated parameters, fitness, and obective functions vaue. 4.3 Genetic operators (1) Seection Seection is to determine which individuas enter the next generation, for which Rouette gambing aw is chosen [2]. (2) Crossing The crossing rate is P c, that is, two individuas cross at a certain ocation, which is simiar to the gene spitting and reorganization.

4 SU Chuaniao et a. / J Transpn Sys Eng & IT, 28, 8(5), 6167 Passenger voume (person/h) Boarding number Aighting number St op Fig. 2 Passenger voumes at different stops (3) Mutation Mutation is that individua in farther popuation overturned at each ocation with certain probabiity P m, namey, from 1 to, or to 1. Mutation can provide possibe soutions by random search in the space and find goba optima soution to a certain extent. 5 umerica exampes 5.1 Parameters choice According to the average income of residents and the working hours, the passengers waiting time vaue is identified (C 1 ) to be.4 yuan/min, and the on board time vaue is.2 yuan/min. With the survey in the bus companies, the operation cost vaue is defined to be.4 yuan/min. With the BRT system in Beiing, Hangzhou, and other cities, the dweing time is 1 min, the acceeration and deceeration time is 4 s, and the running speed is 26 km/h. The number of stations is 12 and the studied period is 1 hour. According to Refs. [1], [2], and [7] and other reevant parameters of genetic agorithm, the crossing rate (P m ) is.8, the mutation rate (P c ) is.5, the popuation size is 2, and times 1. Passengers voume, different forms of bus stops, and the route parameters are shown in Fig. 2 and Tabes 1 and Resut anaysis In Tabe 3, 1 represents norma scheduing, 2 represents zone scheduing, and 3 represents express scheduing. From the resut, with the headway decreased, the obective functions un-optimized decreased graduay because of the decined waiting time. After optimized combination, the headway, the express, and zone scheduing a increased. The combination can aso reduce the tota system cost. From Tabe 3, it can be observed that, after optimization, cost is saved by 69.92% at most and 29.52% at east. The optimization is very significant. The mode and the soution of BRT headway and scheduing combination are feasibe to save system cost greaty. Tabe 1 Scheduing combination of each stop Stop orma schedue Zone schedue Express schedue ote: 1 means stop, means not stop. Stop or not need consider the factors incuding passenger OD, passenger voume, and the and use comprehensivey. Tabe 2 Parameters of BRT route Stop Distance (m) Running time (min) Headway (min) Frequency Tabe 3 Combination of headway and schedue form Combination Obectives vaue Obectives vaue (not optimized) Saving cost (%)

5 SU Chuaniao et a. / J Transpn Sys Eng & IT, 28, 8(5), 6167 Obective vaue (yuan) Trip voume Fig. 3 Sensitivity of obective function to trip voume Obective vaue (yuan) Speed (km/h) Fig. 4 Sensitivity of obective function to speed 5.3 Sensitivity anaysis (1) Impact of traffic voume on obective function To verify the impact of traffic voume on the obective function, the trip voume in Fig. 2 is taken as the standard. Thus, the changes of obective function are examined in the case that trip voume is respectivey.5, 1.5, and 2 times than the appointed standard (Fig. 3). Figure 3 iustrates that with the increase of the traffic voume, trave costs are the argest when the headway is 2 min. Under the given traffic voume, sma headways reduce the obective function vaue, which is mainy caused by different scheduing combinations. There exist much varied soutions, as the frequency decreases and headway increases. Fig. 3 provides the optimum vaue of the obective function under different cases. Despite that, the headway increases, and the system cost does not increase as a whoe. (2) Impact of trave speed on obectives function From Fig. 4, it can be observed that the impact of speed on obective functions is twisted. When the speed reaches 3 km/h, the obective function reduces obviousy. When the speed is km/h, the obective functions with different headways perform differenty, and some vaues increase as speed increases, which indicates that the scheduing combination make an impact on the cost evidenty. Under the same speed, the reasonabe combination of the scheduing can optimize the system cost. 6 Concusions This study focuses on BRT headway and scheduing combination optimization. The mode is deveoped by minimizing passenger trave time and the vehice operation time of BRT. Variabe-ength coding genetic agorithm was used to sove the mode. The obtained optimization resuts indicate that the proposed approach is feasibe. Sensitivity anaysis shows that when the traffic voume increases and the trave speed decreases, a reasonabe aocation of vehices by scheduing form combination woud reduce trave costs. Acknowedgements This research was funded by the ew Century Exceent Researcher Award Program from Ministry of Education of China (CET4946) and the Doctora funding from Ministry of Education of China (25716). References [1] Fan Q S, Pan W. Appication research of genetic agorithm in inteigent transport systems scheduing of vehice. Computer and Digita Engineering, 27, 35(5): [2] Tong G. Appication study of genetic agorithm on bus scheduing. Computer Engineering, 25, 31(13): [3] Dai L G, Liu Z D. Research on the muti-obective assembed optima mode of departing interva on bus dispatch. Journa of Transportation Systems Engineering and Information Technoogy, 27, 7(4): [4] Avishai Ceder. Urban transit scheduing: framework, review and exampes. Journa of urban panning and deveopment, 22, 128(4): [5] Shrivastava P, Dhingra S L. Deveopment of coordinated schedues using genetic agorithms. Journa of Transportation Engineering, 22, 128(1): [6] Ren C X, Zhang H, Fan Y Z. Optimizing dispatching of pubic transit vehices using genetic simuated anneaing agorithm. Journa of System Simuation, 25, 17(9): [7] Teng J, Yang X G. A study on optimizing method for dispatching probem of pubic transit hub under the condition of APTS. Systems Engineering, 24, 22(8): [8] Zou Y. Study on bus regiona scheduing trave pan organizing method. Journa of Transportation Systems Engineering and Information Technoogy, 27, 7(3): [9] Bai Z J, He G G, Zhao S Z, et a. Design and impementation of Tabu search agorithm for optimizing BRT Vehices dispatch. Computer Engineering and Appication, 27, 43(23):